Abstract:

In this paper, we Considered herein the Cauchy problem for a one-parameter family shallow water wave equation which approximate the Euler's equations of motion and the equation of mass conservation in the regime of $\delta\ll 1, \varepsilon={\cal O}(\sqrt{\delta})$. We first establish that this surface equation for shallow water waves of large amplitude is local well-posedness in Sobolev spaces $ H^s(\mathbb{R} )$ with $s>\frac{3}{2}$, which implies that the data-to-solution map is existence, uniqueness and continuous dependence on their initial data, we further show that this dependence is not uniformly continuous in these Sobolev spaces. Moreover, we obtain that the data-to-solution map for this shallow water wave equation is Hölder continuous in the sense of $H^{r}(\mathbb{R} )$-topology for all $0\leq r<s$, and the Hölder exponent $\gamma$ depending on $s$ and $r$. Then, the precise blow-up mechanism for the strong solutions is determined in the Sobolev space $H^{s}$ with $s > 3/2$. In addition, we also investigate the asymptotic behaviors of the strong solutions to this equation at infinity within its lifespan provided the initial data lie in weighted $L_{\phi}^p:= L^{p}(\mathbb{R},\phi^{p}{\rm d}x)$ spaces.

Key words: Shallow water waves, Local well-posedness, Nonuniform continuity, H?lder continuity, Blow up, Persistence property.